Let $S$ be of cardinality $n$ where elements of $S$ are integers from $\mathbb N_{n}$ and at least one element of $S$ is repeated (That is at least one integer from $\mathbb N_{n}$ is skipped. One can easily find a set $S$ with the property that:
$\displaystyle \sum_{j \in S}j^{i} = \displaystyle \sum_{j \in \mathbb N_{n}}j^{i}$
when $i = 1$. (Example: $n=4$, $S=\{1,1,4,4\}$ has sum $10$, the same as sum of first n consecutive integers)

How about for $i \ge 2$? It is not obvious that higher power sum sets exist due to constraint in the cardinality of $S$ and $\mathbb N_{n}$. One cannot deny it either? Is there a easy way to tackle some sumset questions?

For $i=2$ it is related to quadratic forms and integer norms. In an integer coordinate system, how many ways can a given integer norm occur when the coordinates are bounded?

How about if $n \ne 2^{k}$? Is there always such an $S$? Actually I am looking for a negative answer:)? If I have a negative answer there is a way to solve some hard problems in computer science in a somewhat easier manner.
–
TurboOct 17 '10 at 1:29

As soon as you have a sum of distinct $i$th powers, say $a_1^i+\dots+a_s^i$, equal to another sum of (not necessarily distinct) $i$th powers $b_1^i+\dots+b_s^i$ ($s$ is, of course, the same), you have the desired property for $n\ge\max\lbrace a_1,\dots,a_s,b_1,\dots,b_s\rbrace$. So, your question is about a "minimal" solution of
$$a_1^i+\dots+a_s^i=b_1^i+\dots+b_s^i$$
in integers with $a_1,\dots,a_s$. The equation does not look pretty enough, and solutions for small $i$ can be found "by hand".